2210-90 Final Exam
Spring 2014
Name
Instructions.
Show all work and include appropriate explanations when necessary. Correct answers
unaccompanied by work may not receive full credit. Please try to do all all work in the space provided and
circle your final answer.
1. (15pts) Consider the vectors u = 2i − j + k, v = −i + j + 2k, and w = 4i + 2j + k
(a) (3pts) Find u + v − w
(b) (3pts) Find ||w||
(c) (3pts) Find u · v
(d) (2pts) Which two vectors are orthogonal? Circle one:
A. u, v
B. u, w
C. v, w
(e) (4pts) Find u × w
2. (8pts) Find an equation for the plane which contains the points (0, 2, 3), (1, −1, 3) and (1, −2, 1).
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3. (15pts) Consider the function f (x, y) =
p
x2 + y 2 .
(a) (5pts) Find ∇f (x, y), the gradient of f (x, y).
(b) (5pts) Find the equation of the tangent plane to the graph of z = f (x, y) at the point (3, 4, 5).
(c) (5pts) Use part (b) above to estimate the value of
p
(2.9)2 + (4.2)2 .
4. (12pts) Use Lagrange multipliers to find the extreme values (both maximum and minimum) of the
function f (x, y) = x2 + y 2 on the circle (x − 1)2 + y 2 = 1.
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5. (26pts) Evaluate the following double or triple integrals:
ZZ
(a) (5pts)
xy dA, where R is the rectangle 0 ≤ x ≤ 2, 2 ≤ y ≤ 4.
R
ZZ
(x + 2y) dA, where T is the region bounded by the curve y = x − x2 and the x-axis.
(b) (7pts)
T
ZZ
x dA, where S is the piece of the disk x2 + y 2 ≤ 1 in the first quadrant.
(c) (7pts)
S
ZZZ
(d) (7pts)
z
p
x2 + y 2 dV , where C is the cylinder x2 + y 2 ≤ 1, 0 ≤ z ≤ 2.
C
3
6. (10pts) Match the following vector fields with the level curves to which they are perpendicular by
writing the letter in the blank provided.
F(x, y) = xi + yj
F(x, y) = −i + j
F(x, y) = i + j
F(x, y) = xi − yj
F(x, y) = xi
7. (18pts) Consider the vector field F(x, y) = (x2 + y)i + yj.
(a) (4pts) Compute div F
(b) (4pts) Compute curl F
(c) (2pts) Is F a conservative vector field? Circle one:
YES
NO
(d) (6pts) If C1 is the curve parameterized by r(t) = ht, sin (πt)i for 0 ≤ t ≤ 1, compute
Z
F · dr
C1
Z
(e) (2pts) If C2 is another curve connecting (0, 0) to (1, 0), would you expect
Z
to
F · dr? Circle one:
YES
NO
C1
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F · dr to be equal
C2
8. (16pts) Let F(x, y) = (3x + y 3 )i + (y − x3 )j and let C denote the unit circle traversed counterclockwise.
R
(a) (8pts) Compute the line integral C F · dr using Green’s Theorem.
(b) (8pts) Compute the line integral
Plane Divergence Theorem).
R
C
F · n ds using Green’s Theorem (this version is also called the
9. (8pts) Consider the vector field F(x, y, z) = (2xz + y)i + (x2 − 3y)j + (3z − z 2 + y)k. Use the Divergence
Theorem to compute
ZZ
F · n dS,
S
where S is the surface of the ellipse 2x2 + y 2 + z 2 = 1, and n is the outward pointing unit normal
vector.
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10. (6pts) Let C be the curve and vector field given by the picture below. Indicate whether each statement
is true (T) or false (F) by writing in the blank provided.
R
F · dr < 0
RC
F · n ds = 0
C
F is conservative.
11. (16pts) Let S denote the surface determined by
z = x2 − y 2 ,
where x2 + y 2 ≤ 1 (S is the graph of f (x, y) = x2 − y 2 over the unit disk).
(a) (8pts) Find the surface area of S.
(b) (8pts) Use Stokes Theorem to evaluate
ZZ
(curl F) · n dS,
S
where F denotes the vector field
F = (x2 + y 2 − 1)i + (z − x2 + y 2 )k.
Hint: What does F look like on the boundary of S (the graph of z = x2 − y 2 over the circle
x2 + y 2 = 1)?
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